In mathematics, the Dynkin index
of finite-dimensional highest-weight representations of a compact simple Lie algebra
relates their trace forms via
is the highest root, so that
is the adjoint representation, the Dynkin index
is equal to the dual Coxeter number.
is the trace form on the representation
By Schur's lemma, since the trace forms are all invariant forms, they are related by constants, so the index is well-defined.
Since the trace forms are bilinear forms, we can take traces to obtain[citation needed] where the Weyl vector is equal to half of the sum of all the positive roots of
is the value of the quadratic Casimir in the representation